Expansion of the universe

I wonder if I am the only person who has found the various explanations available in the public domain, of the expansion of the universe, confusing. For about two years after learning the essentials of general relativity I struggled with the apparent contradictions in the idea of cosmological expansion. I asked several questions on internet forums but never got an answer that helped me resolve the confusion. That was probably because I was asking the wrong questions, but it is so hard to know what are the right questions to ask.

I am fortunate to have finally been able to discover the right questions and, together with answers to those, and some useful papers by cosmologists, to piece together a coherent understanding of what is actually going on. This essay is an attempt to explain that for anybody else who finds the concepts puzzling.

I’ll start by describing why the concept of an expanding universe is confusing. The usual explanation involves an analogy with a balloon that is being inflated, on which dots have been marked with a felt pen. As the balloon inflates, the dots – which correspond to galaxies – become further apart. There are three key problems with this analogy, and with the idea of expansion in general:

The balloon analogy fails because it seems to require the existence of a physical object that is space, corresponding to the rubber of the balloon. But the Michelson Morley experiments showed that there is no ‘absolute space’, which was referred to at the time as the ‘luminiferous ether’. The balloon analogy implies the existence of a ‘privileged reference frame’, and they are not supposed to exist.

We are told that faraway galaxies are receding from us faster than the speed of light. This appears to contradict the rule that nothing can travel faster than light.

If the universe’s expansion is accelerating then that seems to imply that energy is being created, as the kinetic energy of the galaxies will be ever increasing, without any compensatory reduction in potential energy.

The Cosmological Principle – a Privileged Reference Frame

To start on trying to resolve these objections, we need to first state the fundamental principle on which this sort of analysis is based – the Cosmological Principle. This principle effectively says that there exists a system of coordinates for spacetime under which a snapshot of the universe at any ‘cosmic’ time coordinate is isotropic and homogeneous at the large scale.

A system of coordinates (also known as a ‘coordinate system‘ or ‘reference frame‘) is a scheme that assigns a unique set of four numbers to every point in spacetime. This enables any point in spacetime to be referred to by those numbers – its ‘coordinates’. One of those coordinates is a time coordinate and the other three are spatial – denoting a location in space.

Homogeneous means it is the same everywhere. This rules out possibilities such as there being a ‘centre of the universe’ where stars occur most frequently, with the distribution of stars getting ever sparser as you travel further from that centre.

Isotropic means it is the same in every direction. So an observer that is stationary with respect to this coordinate system would not for instance observe stars on her left moving towards her on average and stars on her right moving away from her.

At the large scale means we ignore local variations. Of course stars are more frequent within a galaxy than outside one, but once we zoom our perspective out enough to incorporate many clusters of galaxies, the density of matter should appear roughly constant. Similarly, we may have more objects within our galaxy moving towards us from the left than from the right – meaning local isotropy does not hold – but if we zoom out enough and take enough distant galaxies into our view, their motion relative to us will be the same in all directions.

The homogeneity assumption is easy enough to grasp. It simply says there is nothing special about our part of the universe, that what we can see is a fair sample of the whole thing.

The isotropy assumption is the really powerful one, because it establishes the privileged reference frame that we can associate with the rubber in the balloon analogy. Given any point P in spacetime, we can define a location in space as being the unique worldline through P for which the universe appears isotropic from every point in that worldline. In the analogy, that means that the worldline follows the path of a dot marked on the balloon’s surface, as it is being blown up.

By the way, an isotropic universe must be homogeneous, but the reverse is not true. A homogeneous but non-isotropic universe could be one in which the speed of light is different depending on which direction the light is travelling.

We can now satisfy the first objection. There is indeed a privileged reference frame, being the one that generates the required conditions of homogeneity and isotropy. It does not require the existence of a substance like the luminiferous ether, whose existence was disproven by the Michelson Morley experiments. All it requires is the existence of matter throughout the universe, and the application of the Cosmological Principle. The existence of this frame does not contradict the principles of Galilean and Special Relativity. Those principles state that there is no privileged ‘inertial reference frame’, which means a frame that is not accelerated or subject to gravitational forces. The frame identified by the Cosmological Principle is not inertial and hence does not contradict those theories. In one sense it is similar to the ‘laboratory frame’ on Earth, which is the frame in which the laboratory in which experiments are being conducted is stationary. That frame is privileged in a sense, but like the cosmological frame it is not inertial, because of the effect of the Earth’s gravity.

The spatial coordinates of a point that is stationary with respect to the cosmic reference frame are frequently called ‘co-moving coordinates’. In the balloon analogy this means that a point with those coordinates is moving in the same way as the rubber, i.e. it is stationary with respect to the rubber.

Given this special reference frame, we can now make sense of another commonly used aspect of the balloon analogy, that of ants crawling on the balloon. In the balloon analogy, there is a clear difference between the relative motion of dots marked on the balloon’s surface and the relative motion of ants crawling on that surface. In the former case, the motion is caused solely by the expansion, deflation or other deformation of the balloon. In the latter, the motion is a combination of that with the motion of the ant relative to the rubber.

We can use the same concept in our universe model. The coordinates of a stationary section of the ‘rubber’ are determined by application of the cosmological principle. Objects that are stationary in those coordinates may still be moving relative to one another due to the expansion of the universe. On top of that, an object may be moving locally, relative to that coordinate system. For instance, the Earth is orbiting the sun, which involves constant changes of direction, incorporating a full 180 degree change of direction of motion every six months. This motion is relative to the cosmic coordinates, and we call it ‘peculiar motion’. It is analogous to an ant walking around in circles on the rubber of the balloon.

The Andromeda galaxy is moving towards ours under gravitational attraction, and they seem destined to collide. Those relative motions are also peculiar motion, like two ants rushing towards one another along the surface of an expanding balloon.

One final note on this special cosmic reference frame. The standard way to determine it is by reference to the Cosmic Microwave Background Radiation (CMBR), which is the radiation left over from the Big Bang, that fills the sky in every direction. We can identify the co-moving coordinates of our location by identifying the velocity we would need to have for the wavelength of the CMBR to be the same in every direction. An observer that is not ‘co-moving’ with the cosmic reference frame will see longer wavelengths in one direction than in its opposite, because of the Doppler effect.

But how can galaxies travel faster than light?

The resolution to the second objection comes from the realisation that the prohibition on faster than light (‘superluminal’) travel, precisely stated, is not the same as how it is often popularly described. The popular description is either:

an object cannot travel faster than light, or

two objects cannot have a relative speed that exceeds the speed of light

Neither of these is strictly correct. They should be really stated as:

1a. There is no inertial reference frame in which an object’s velocity is faster than light, and

2a. Two objects within the same inertial reference frame cannot have a relative speed that exceeds the speed of light

We can immediately see that 2a is not breached by the motion of distant galaxies, because there is no inertial reference frame that contains both us and such a distant galaxy. A reference frame is inertial if spacetime is very close to flat within that frame, and there is too much spacetime curvature between us and a distant galaxy for any reference frame containing both to be flat. This can be understood by comparison to the Earth. It is reasonable to assume the Earth is flat in measuring travelling distances between my house and the local shops. However, if I am planning a trip between from my house in Sydney to London, I have to take the curvature of the Earth into account.

1a also is not breached by the distant galaxy, because it is not travelling faster than light in any inertial frame containing it. Nor is it travelling faster than light relative to any object close enough to it to be in the same inertial frame.

This may seem all a little unsatisfactory, as it leaves a big grey area in between the local shops and the distant galaxy, in which we are uncertain how far we can extend an inertial reference frame. Fortunately, we can resolve this by expressing the prohibition on superluminal travel in a more precise way, as follows:

No object can have a spacelike four-velocity.

A four-velocity is a vector that can be used to represent an object’s motion, which is independent of any reference frame (‘co-ordinate independent’). Like all vectors, it has a magnitude (‘size’) and a direction, but the direction is in spacetime, not space. In any given reference frame, a four-velocity can be denoted by four numbers, called components, of which one will be a ‘time’ component and the other three will be ‘spatial’ components. The values of the four components will differ between reference frames, but they will all refer to the same physical phenomenon, whose magnitude and direction does not differ between reference frames. There is a mathematical formula, involving something called the metric tensor, for determining the magnitude of any vector. In relativity, magnitudes can be positive, negative or zero. A vector with negative magnitude can be the velocity of an object with mass, and such vectors are called ‘timelike‘. Light rays, which have no mass, must have velocity vectors with zero magnitudes, and those vectors are called ‘lightlike‘*. Vectors with positive magnitudes are called ‘spacelike‘.

The law 3 is the most general and precise statement of the prohibition on superluminal travel. Prohibitions 1a and 2a are consequences of this rule and, given a local, inertial reference frame, prohibitions 1 and 2 follow.

In an expanding universe, no object, including no galaxy, can have a spacelike velocity, so the prohibition is respected. It will be the case that the distance between two galaxies far away from one another is increasing at more than 3 x 108 ms-1, but that does not breach the prohibition.

One last point on this objection. When we refer to distance in that last paragraph, we mean what is called the ‘proper distance’. That means the shortest distance between the two galaxies in the snapshot of the universe taken at an instant of cosmic time. In an expanding universe that shortest distance will be increasing with cosmic time.

Does the expansion violate conservation of energy?

If galaxies are rushing apart at ever increasing rates that appears to increase the net energy of the universe on two counts. Firstly, their kinetic energy is increasing with their increasing relative velocities. Secondly, their gravitational potential energy is also increasing as they get farther apart. This appears to violate the principle of conservation of energy.

The answer to this objection is that conservation of energy only applies locally, within an inertial reference frame. There is no coordinate-dependent global equivalent to that local principle. In fact there is not even any global, coordinate-independent definition of total energy. There are approaches using coordinate-dependent pseudo-tensors, but these are controversial, and seem unsatisfying given their coordinate-dependence.

Even so, it appears that these pseudo-tensor approaches can be used to derive a conclusion that the net energy of the universe is zero, and will remain zero. Hence, under such an analysis, there does not appear to be any violation of conservation of energy.

* Note: If you are familiar with vectors in two and three-dimensional Euclidean space, you might assume that any vector with zero magnitude must be a trivial vector whose components are all zero in any reference frame, and hence which has no direction. While that conclusion is true for Euclidean space, it is not true for spacetime, which is non-Euclidean. Lightlike vectors have zero magnitudes but they also have a well-defined direction. They will have non-zero components, which cancel one another out when we calculate the magnitude.